Dual space

Dual space

The dual space is a vector space that is defined by the natural definitions of addition and multiplication by a scalar. For finite-dimensional vector spaces, the dual space is the space of linear functionals. It is the dual space of a space of linear functionals over a given vector space. The dual space of a finite-dimensional space has the same dimension as the original space. When given a finite-dimensional real vector space, its dual space is denoted by the symbol "V^*".From: Mathematical Methods for Physics and Engineering [2018], The Limits of Resolution [2019], Blow-up time analysis of parabolic equations with variable nonlinearities [2022], Advanced Linear Algebra [2019], Tensor methods for finding approximate stationary points of convex functions [2022]

Linear Algebra

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Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012

Fatemeh Hamidi Sepehr, Erchin Serpedin

forms a unique basis (termed the dual basis of B) for the dual spaceV*.

Blow-up time analysis of parabolic equations with variable nonlinearities

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Published in Applicable Analysis, 2022

Benkouider Soufiane, Rahmoune Abita

We say that for a function, is a weak solution of problem (1) if for every test-function and every , the following identity holds: where is the dual space of (the space of linear functionals over ). For every fixed we introduce the Banach space with the associate norm and denote by its dual.

Quasiconvex optimization problems and asymptotic analysis in Banach spaces

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Published in Optimization, 2020

A. Iusem, F. Lara

Let V be a reflexive Banach space with norm . By we mean the duality pairing of two elements from V and , where is the dual space of V. Given a nonempty set , we denote its strong closure by , its boundary by , its topological interior by and its relative interior by . By we mean the closed ball with centre at and radius . By ‘’ we mean weak convergence while ‘’ means strong convergence.

Optimality conditions for the simple convex bilevel programming problem in Banach spaces

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Published in Optimization, 2018

Susanne Franke, Patrick Mehlitz, Maria Pilecka

The (topological) dual space of is denoted by , and is the corresponding dual pairing. Let be nonempty. Then denotes the weak--closure of B, i.e. the closure with respect to (w.r.t.) the weak--topology in .

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Linear Algebra

Blow-up time analysis of parabolic equations with variable nonlinearities

Quasiconvex optimization problems and asymptotic analysis in Banach spaces

Optimality conditions for the simple convex bilevel programming problem in Banach spaces